Method for providing error information relating to inconsistencies in a system of differential equations

ABSTRACT

The method and the computer-related products provide for error information relating to inconsistencies in a system of differential equations that describes a technical system or a technical process.

CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application is a continuation of copending InternationalApplication No. PCT/DE02/01878, filed May 23, 2002, which designated theUnited States and which was not published in English.

BACKGROUND OF THE INVENTION FIELD OF THE INVENTION

[0002] The invention relates to a method for providing error informationrelating to inconsistencies in a system of differential equations.

[0003] It is known in the prior art to describe a technical system withsystems of differential equations. With the assistance of numericalmethods using a computer system, or of an analog computer, it ispossible to simulate the systems by calculating solutions to the systemsof differential equations for specific initial or boundary conditions.Examples for the description of technical systems and typical simulationmethods can be taken from standard works of the relevant field ofspecialization. Detailed instructions therefor are specified, forexample, in G. Schmidt, “Grundlagen der Regelungstechnik” [“Principlesof Control Engineering”], 2nd edition, Springer, Berlin, 1987, inUnbehauen, “Regelungstechnik I” [“Control Engineering I”], 6th edition,Friedr. Viehweg & Sohn, Braunschweig/Wiesbaden, 1989, in Unbehauen,“Regelungstechnik II” [“Control Engineering II”], 5th edition, Friedr.Viehweg & Sohn, Braunschweig/Wiesbaden, 1989, in Unbehauen,“Regelungstechnik III” [“Control Engineering III”], 2nd edition, Friedr.Viehweg & Sohn, Braunschweig/Wiesbaden, 1986, in E. Pfeiffer,“Einführung in die Dynamik” [“Introduction to Dynamics”], B. G. Teubner,Stuttgart, 1989, and in E. Ziegler (ed.), “Teubner-Taschenbuch derMathematik” [“Teubner Manual of Mathematics”] B. G. Teubner, Stuttgart,1996. When simulating, the problem frequently arises that a method fornumerically solving a system of differential equations will terminatebecause the fundamental system of differential equations is singular.For the same reason, it can happen that an analog computer cannotforesee the system behavior. In addition, it frequently happens thatsolutions that are not plausible occur when solving the systems ofdifferential equations.

[0004] In such a case, it is known in the prior art that the programmeror a user of a program for solving differential equations is responsiblefor locating errors. With extensive experience, it is possible to locatethe causes of the singularity and, if appropriate, to reduce errors inthe modeling of the technical system. This is generally very timeconsuming and costly.

[0005] A method for automatic structure analysis of very large systemsof differential-algebraic equations is described in S. Chowdry et al:“Automatic structure analysis of large scale differential algebraicsystems” IMTC 2001. Proceedings of the 18th IEEE Instrumentation andMeasurement Technology Conference, Budapest, Hungary, May 21-23, 2001,IEEE Instrumentation and Measurement Technology Conference (IMTC): NewYork, N.Y.: IEEE, US, Vol. 1 of 3, Conf. 18, May 21, 2001 (2001-05-21),pages 798-803. This method takes account both of numerical and ofsymbolic information, and also incorporates the effects of linear termsrigorously into the mathematical system, and is capable of providing avery accurate representation of the structure of a system ofdifferential-algebraic equations.

[0006] A structural algorithm that takes account of the structuralproperties of systems of differential-algebraic equations is describedin the scientific article by J. Unger et al: “Structural analysis ofdifferential-algebraic equation systems-theory and applications” Comput.Chem. Eng.; Computers & Chemical Engineering August 1995, Pergamon PressInc., Tarrytown, N.Y., USA, Vol. 19, No. 8, August 1995 (1995-08), pages867-882. In that case, two systematically different structuralapproaches are implemented on a computer system in order to analyzesystems of differential-algebraic equations structurally. These computerprograms are applied both as separate programs for supporting thedevelopment of simulation models with a low index, and as constituentsof the DIVA simulation environment, in order to facilitate a selectionof suitable numerical algorithms, and to calculate consistent initialconditions.

[0007] The fact that the structural index of a lineardifferential-algebraic equation with constant coefficients and with theindex 1 can be arbitrarily high and to that extent contradicts a resultpreviously published in the literature is shown by the scientificpublication by Gunter Reissig et al: “Differential-algebraic equationsof index 1 may have an arbitrarily high structural index” Siam J. SCI.Comput.; Siam Journal of Scientific Computing 2000 SOC for Industrial &Applied Mathematics Publ., Philadelphia, Pa., USA, Vol. 21, No. 6, 2000,pages 1987-1990. This publication further states that when applied todifferential-algebraic equations with an index of 1, Pantelides'algorithm can execute an arbitrarily high number of iterations anddifferentiations.

SUMMARY OF THE INVENTION

[0008] It is accordingly an object of the invention to provide a methodfor providing error information relating to inconsistencies in a systemof differential equations which overcomes the above-mentioneddisadvantages of the heretofore-known devices and methods of thisgeneral type.

[0009] With the foregoing and other objects in view there is provided,in accordance with the invention, a method for predicting the behaviorof a technical system starting from prescribed system properties,boundary conditions, and/or starting from prescribed influences on thesystem. The method comprises the following steps:

[0010] providing a computer system;

[0011] defining a system of equations describing the technical system,the equations having the form f(t,x(t), {dot over (x)}(t), . . . , x^((k))(t),p)=0, and including: $\begin{matrix}{{{f_{1}\left( {t,{\underset{\_}{x}(t)},{\underset{\_}{\overset{.}{x}}(t)},\ldots \quad,{{\underset{\_}{x}}^{(k)}(t)},\underset{\_}{p}} \right)} = 0},} \\{{{f_{2}\left( {t,{\underset{\_}{x}(t)},{\underset{\_}{\overset{.}{x}}(t)},\ldots \quad,{{\underset{\_}{x}}^{(k)}(t)},\underset{\_}{p}} \right)} \eqsim 0},} \\\vdots \\{{{f_{n}\left( {t,{\underset{\_}{x}(t)},{\underset{\_}{\overset{.}{x}}(t)},\ldots \quad,{{\underset{\_}{x}}^{(k)}(t)},\underset{\_}{p}} \right)} = 0},}\end{matrix}$

[0012] wherein x(t) and derivatives {dot over (x)}(t), . . . , x^((k))thereof in each case have m elements, and p is a parameter vectorthat can occur in the system of equations; and

[0013] executing a test method for providing error information relatingto inconsistencies in the system of equations on the computer system,the test method having the following steps 1 to 3:

[0014] step 1: setting up a dependence matrix A with m columns and nrows, and setting an element A to A(i,j)≠0 when an i^(th) row of fdefined with f _(i)(t,x(t), {dot over (x)}(t), . . . , x ^((k))(t),) isa function of

[0015] a) a j^(th) element of x expressed as x _(j)(t); or

[0016] b) one of the derivatives of the j^(th) element of x defined as x_(j) ^((s))(t);

[0017] and otherwise setting the element A(i, j)=0;

[0018] step 2: determining a set of row ranks each having the numbers ofthose rows of the dependence matrix A that are mutually dependent anddetermining a set of column ranks of the dependence matrix A each havingthe numbers of those columns of the dependence matrix A that aremutually dependent if such row ranks and/or column ranks are present;and

[0019] step 3: outputting error information containing the numberscontained in each row rank determined in step 2 and in each column rankdetermined in step 2.

[0020] The invention is based on the idea of examining the system ofdifferential equations for inconsistencies with the aid of combinatorialmethods as early as before a simulation, that is to say before executingnumerical calculating steps or building an analog computer. Wheninformation relating to such inconsistencies is at hand, it is possibleto deduce errors in the modeling of the technical system, or errors inthe technical system itself, that can cause undesired contamination in alater simulation step, or even render the system incapable of beingsimulated. There is no need for a time-consuming execution of thesimulation as far as termination, nor for a trial simulation. Thecorrection of the model of the technical system or of the technicalsystem itself is substantially facilitated by the presence ofinformation relating to inconsistencies.

[0021] The first step, of the method according to the invention,provides for setting up a dependence matrix A. The dependence matrix Ahas exactly as many columns as the dimension of the solution vector x ofthe system of differential equations. The dependence matrix A hasexactly as many rows as the number of differential equations in thegiven system of differential equations.

[0022] The invention is not limited in this case only to systems ofdifferential equations of the form f(t,x(t),{dot over (x)}(t), . . . , x^((k))(t),p)=0. It can also be applied to special cases in which thesystem of equations for simulating the technical system has the form,for example, of f(t,x(t))=0. In such a case, the talk is not of a systemof differential equations but of a generally nonlinear system ofequations.

[0023] Moreover, it is not absolutely necessary for the system ofdifferential equations that is provided for the simulation to beexamined itself for inconsistencies. Moreover, another system ofdifferential equations that describes the same technical system or thesame technical process can be examined for inconsistencies. This can beadvantageous when the last-named system of differential equations has asimpler structure than that to be simulated later. For example, in thesimulation of electric networks the equations of modified node voltageanalysis are usually the basis of the numerical simulation, although theso-called branch voltage-branch current equations have a substantiallysimpler structure. It is even possible frequently to determine moreinconsistencies or to limit inconsistencies more severely by examiningsystems of differential equations with a simple structure.

[0024] The dependence matrix A in this case has elements A(i,j) that areset either to the value “zero” or to the value “nonzero”, without regardto the absolute values of the elements of A. In this case, the elementA(i,j) can be set to a value ≠ 0 when the i^(th) row of f is a functionof the j^(th) element of x, or of one of the derivatives of the j^(th)element of x. In all other cases, the element A(i,j) is set to the value“0”.

[0025] It is immaterial for the method according to the inventionwhether the dependence matrix A is actually set up explicitly. Rather,what is important is the corresponding information as to whether thei^(th) row of f is a function of the i^(th) element of x or of one ofits derivatives, or not, being available in the following method steps.

[0026] In the second step of the method according to the invention, aset of row ranks is determined that in each case have the numbers ofthose rows of the dependence matrix A or those rows of the system ofdifferential equations that are mutually dependent. Moreover, a set ofcolumn ranks of the dependence matrix A is determined that in each casehave the numbers of those columns of the dependence matrix A or thosecomponents of x or one of its derivatives that are mutually dependent.

[0027] Here, “row rank” is understood as follows. A set C_(z) of naturalnumbers i, 1≦i≦n, is a row rank of a matrix with n rows and m columnswhen it fulfils the following conditions:

[0028] (i) there is no transversal T of the matrix A such that C_(z) iscontained in the set of row indices of T.

[0029] (ii) for each element c of C_(z) there is a transversal T of Asuch that C_(z)\{c} is wholly contained in the set of the row indices ofT.

[0030] Here, the expression C_(z)\{c} represents that set which resultswhen the element c is removed from the set C_(z).

[0031] A transversal T of the matrix A is understood as follows. Atransversal of a matrix A with n rows and m columns is one of, ifappropriate, a plurality of possible sets of positions (i, j) ofnon-vanishing matrix entries A (i, j), of which no two or more are inthe same row or column. A set T of pairs (i, j), 1≦i≦n,1≦j≦m,constitutes a transversal of the matrix A if T fulfils the followingconditions:

[0032] (i) A (i, j) does not vanish for all elements (i, j) of T, and

[0033] (ii) given that (i, j) and (i′, j′) are two different elements ofT, that is to say i≠i′ or j≠j′, it holds that i≠i′ and at the same timej≠j′.

[0034] It is of less importance for the invention in this case whetherone of the possible transversals of the dependence matrix A iscalculated. The definition of the term “transversal” is required here,in order to illustrate the terms “row rank” and “column rank”.

[0035] The term “row index of the transversal T” is understood asfollows. The set Z of the “row indices of the transversal T” has aselements the row indices of the elements of T. In other words, thismeans that the set Z contains precisely those numbers i for which thereis a j such that the element (i, j) is an element of T.

[0036] A column rank is understood as a set C_(s) of natural numbers i,1≦i≦m, of a matrix A with n rows and m columns when it fulfils thefollowing conditions:

[0037] (i) there is no transversal T of the matrix A such that C_(s) iscontained in the set of column indices of T.

[0038] (ii) for each element c of C_(s) there is a transversal T of thematrix A such that C_(s)\{c} is wholly contained in the set of thecolumn indices of T.

[0039] The “set of column indices of the transversal T” is understood asa set Z that contains as elements the column indices of the elements ofthe transversal T of the matrix A. In other words, this means that theset Z contains precisely those numbers j for which there is an i suchthat the element (i, j) is an element of the transversal T of the matrixA.

[0040] After step 2 according to the invention is carried out, there ispresent a set of row ranks that respectively have the numbers of thoserows of the dependence matrix A or those rows of the system ofdifferential equations that are mutually dependent. Moreover, a set ofcolumn ranks of the dependence matrix A is present that in each casehave the numbers of those columns of the dependence matrix A or of thosecomponents of x or one of its derivatives that are mutually dependent.

[0041] It is particularly easy having this information to deducestructural errors of the system of differential equations that ispresent as starting point. Such structural errors or inconsistencies inthe original system of differential equations are frequently causes oferrors in the calculation of the solution of the system of differentialequations.

[0042] A basic step in executing the method according to the inventionconsists in finding row ranks and column ranks. Methods for finding rowranks and column ranks are known. Methods for determining “row ranks”are to be found in the literature under the keyword of “minimallystructurally singular subsets of equations” (compare C. C. Pantelides,The consistent initialization of differential-algebraic systems in SIAMJ. Sci. Statist. Comput., 9(2):213-231, March 1988). In order todetermine “column ranks”, it is possible to use a method for determining“row ranks” when a transposed matrix is determined from that matrix ofwhich the “column ranks” are to be determined. The row ranks of thecorresponding transposed matrix are then determined in order todetermine the column ranks of the matrix.

[0043] This manner of determining the row ranks and column ranks for themethod according to the invention is not to be understood in a limitingway. Rather, it is also possible to use other methods to determine rowranks and column ranks if the remaining conditions for the presence ofrow ranks and column ranks are fulfilled.

[0044] In the concluding step of the method according to the invention,for each row rank and for each column rank that had been determined inaccordance with step 2, the numbers that are contained therein areoutput. In the case of the row ranks, these numbers indicate the runningnumbers of the equations of the system of differential equations thatare possibly affected by a structural problem. In the case of columnranks, the numbers contained therein and output as error informationindicate the numbers of the components of the solution vector x that arepossibly affected by a structural problem.

[0045] The method according to the invention can be used even before theexecution of numerical calculating steps or the building of an analogcomputer to check the structure of the system of differential equationsthat is to be calculated for its consistency. If the method according tothe invention had detected a system of differential equations as beinginconsistent, it is possible to locate a large number of the possiblestructural errors of the system of differential equations. Thisaccelerates the location of errors. In addition, time-consumingsimulation tests are avoided.

[0046] The method according to the invention can be applied for any typeof a simulation system for the numerical solution of systems ofdifferential equations, and for the solution of systems of differentialequations by means of an analog computer.

[0047] In a particularly advantageous development of the invention,before executing step 1 of the method according to the invention anequation significance list of length n is applied in which each equationof the system of equations is assigned an equation number and/or an itemof equation text information. In exactly the same fashion, beforeexecuting step 1 of the method according to the invention, a componentsignificance list of length m is applied in which each component of thesolution vector x is assigned a component number and/or an item ofcomponent text information. In this case, the equation text informationor component text information respectively stored in the significancelist is advantageously selected such that this information acquires asignificance in connection to the technical system to be simulated.Consequently, constituents of the simulated technical system andsubstructures of the technical system to be simulated can be assignedequations of the system of differential equations and components of thesolution vector x that facilitate an interpretation of the errorinformation output with the aid of the method according to theinvention.

[0048] In step 3 of the method according to the invention, for thispurpose it is provided to output the equation number and/or the item ofequation text information in accordance with the equation significancelist instead of outputting the numbers contained in each row rank. Forthe case in which a whole number i is contained in the row rank, it isnot the number i that is output but rather the contents of the i^(th)component of the equation significance list. In exactly the samefashion, in step 3 of the method according to the invention thecomponent number and/or the item of component text information inaccordance with the component significance list is/are output instead ofoutputting the numbers contained in each column rank. For the case inwhich the whole number j is present in the column rank as a number, itis not the number j that is output, but rather the contents of thej^(th) component of the component significance list.

[0049] Owing to this refinement of the method according to theinvention, significance contents that refer directly to the technicalsystem to be simulated are output as error information. This errorinformation can be used to illustrate with particular ease, systematicerrors in the structure of the system of differential equations fordescribing the technical system, such that locating the errors isfurther accelerated.

[0050] In the method according to the invention a preferably digitalcomputer is used that has at least a memory, an arithmetic unit, aninput device, and an output device.

[0051] The invention is also implemented in a computer program forproviding error information relating to inconsistencies in a system ofequations. The computer program is designed in this case such that afterinputting of the system properties, the initial or boundary conditionsand the influences on the system, it is possible to execute an inventivemethod in accordance with one of the preceding claims. It is possible inthis case to output results of a simulation, the solution vector or thesolution vectors at different points in time. However, it is alsopossible for the computer program to provide only information relatingto inconsistencies in accordance with one of the preceding claims.

[0052] Because numerous defective program runs and/or simulation trialscan be avoided, the computer program according to the invention resultsin substantial improvements in run time by comparison with the knownsimulation programs and simulation methods.

[0053] Moreover, the invention relates to a data carrier with such acomputer program, and to a method in which such a computer program isdownloaded from an electronic data network, such as from the Internetonto a computer connected to the data network.

[0054] The computer system according to the invention is designed suchthat it is possible to execute on it a method according to the inventionfor providing error information relating to inconsistencies in a systemof equations.

[0055] Finally, the invention also relates to the use of a methodaccording to the invention and/or a computer system for providing errorinformation relating to inconsistencies in a system of equations.

[0056] In the method according to the invention, the first step is toset up the system or system of differential equations that describes theprocess, or other forms such as, for example, nonlinear systems ofequations that have no differential equations. It is assumed here, inthis case, that the person skilled in the art is conversant with themode of procedure required for this purpose. This is examined in detailin the specialist literature mentioned in the introduction to thedescription. There are also relevant computer programs for this purpose.

[0057] Other features which are considered as characteristic for theinvention are set forth in the appended claims.

[0058] Although the invention is illustrated and described herein, ingeneral, as embodied in a method for providing error informationrelating to inconsistencies in a system of differential equations, it isnevertheless not intended to be limited to the details shown, sincevarious modifications and structural changes may be made therein withoutdeparting from the spirit of the invention and within the scope andrange of equivalents of the claims.

[0059] The construction and method of operation of the invention,however, together with additional objects and advantages thereof will bebest understood from the following description of specific embodimentswhen read in connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0060]FIG. 1 is a set of functions and a parameter vector illustrating afirst exemplary embodiment of the invention;

[0061]FIG. 2 is a matrix and a corresponding set of transversalelements;

[0062]FIG. 3 is a set of functions and a parameter vector describing asecond exemplary system;

[0063]FIG. 4 is a matrix, a transversal, and column and row results;

[0064]FIG. 5 is an exemplary display of error results;

[0065]FIG. 6 is a schematic of an equivalent circuit;

[0066]FIG. 7 shows an equation significance list G and a componentsignificance list K; and

[0067]FIG. 8 is a resultant error list.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0068] The first exemplary embodiment in accordance with the invention,relates to the simulation of a technical system that can be described bythe numerical solution of the system of equations

f (t, x (t), {dot over (x)} (t), . . . , x ^((k))(t), p )=0

[0069] of the form

f ₁(x ₁(t),x ₂ (t),x ₃ (t),{dot over (x)} ₁,(t),{dot over (x)} ₂(t),{dotover (x)} ₃(t),p ₁ ,p ₂ ,p ₃)=0,

f ₂(x ₁(t),x ₂ (t),x ₃ (t),{dot over (x)} ₁,(t),{dot over (x)} ₂(t),{dotover (x)} ₃(t),p ₁ ,p ₂ ,p ₃)=0,

f ₃(x ₁(t),x ₂ (t),x ₃ (t),{dot over (x)} ₁,(t),{dot over (x)} ₂(t),{dotover (x)} ₃(t),p ₁ ,p ₂ ,p ₃)=0,

[0070] having the functions f1, f2 and f3, and having the parametervector p in accordance with FIG. 1.

[0071] In order to predict the behavior of the system, the system ofequations is solved numerically, that is to say values for the unknownvector x(t) are calculated at one or more points in time t. For thispurpose, known methods of numerical solution are used which run as acomputer program on a computer system.

[0072] According to the invention, steps for providing error informationrelating to inconsistencies in the system of equations are executedbefore the actual solution of the system of equations.

[0073] The dependence matrix A specified in FIG. 2 is determined in step1 of the method according to the invention, all non-vanishing elementsof the dependence matrix A being denoted in FIG. 2 by a star (“*”)

[0074] The element A(1,1) picked out by way of example is set to anessentially arbitrary non-vanishing value “*”, because the first elementof f, that is to say f₁ (t,x(t),{dot over (x)}(t), . . . , x ^((k))(t)),is a function of the first element of x, that is to say of x ₁(t). Theelement A(1,3) picked out by way of example is set to the value “0”,because the third element of f, that is to say f₃ (t,x(t),{dot over(x)}(t), . . . , x ^((k))(t)), is independent of the first element of x,that is to say of x ₁(t) and of the derivatives of the first element ofx, that is to say x ₁ ^((S))(t).

[0075] In step 2 of the method according to the invention, the firststep is to determine the transversals T, specified in FIG. 2, of thedependence matrix A with the aim of determining row ranks and columnranks of the dependence matrix A specified in FIG. 2. Methods fordetermining transversals (of “maximum cardinality bipartite matchings”)are specified in I. S. Duff, “On algorithms for obtaining a maximumtransversal”, ACM Trans. Math. Software, 7(3):315-330, 1981, in E. L.Lawler, “Combinatorial Optimization: Networks and Matroids”, Holt,Rinehart and Winston, 1976, in L. Lovasz and M. D. Plummer, “MatchingTheory”, North-Holland Mathematics Studies 121. Annals of DiscreteMathematics, 29. North-Holland, 1986, or in C. C. Pantelides, “Theconsistent initialization of differential algebraic systems”, SIAM J.Sci. Statist. Comput., 9(2):213-231, March 1988.

[0076] The transversal T from FIG. 2 has 3 elements. Since n=m=3, thedependence matrix A from FIG. 2 has neither row ranks nor column ranks.No such row ranks of the dependence matrix A are found by the method,specified in C. C. Pantelides “The consistent initialization ofdifferential algebraic systems”, for calculating row ranks (“minimallystructurally singular subsets of equations”). The method specified therefor calculating row ranks also finds no such row rank of the transposeof the dependence matrix A. It follows that there is no column rank ofthe dependence matrix A. The set, determined in step 2 of row ranks ofthe dependence matrix A is empty. The set, determined in step 2 ofcolumn ranks of the dependence matrix A is likewise empty. Consequently,no error information is output in step 3 of the method according to theinvention.

[0077] The system of equations specified at the beginning can be solvedwith a higher probability without the occurrence of errors. If thesimulation is terminated nevertheless, or if it supplies implausiblesolutions, or if it is entirely impossible, then it is possible toexclude structural errors in the description of the technical system orin the technical system itself with a high degree of probability. Thissubstantially simplifies the location of errors.

[0078] The second exemplary embodiment in accordance with the inventionrelates to the simulation of a further technical system (not shownhere), whose behavior can be described by the numerical solution of thesystem of equations

f (t,x (t), {dot over (x)} (t), . . . , x ^((k))(t), p )=0,

[0079] of the form

f ₁(x ₁(t),x ₂(t),x ₃(t), p ₁ , p ₂ , p ₃)=0

f ₂(x ₁(t),x ₂(t),x ₃(t), p ₁ , p ₂ , p ₃)=0

f ₃(x ₁(t),x ₂(t),x ₃(t), p ₁ , p ₂ , p ₃)=0

[0080] having the functions f1, f2 and f3 and having the parametervector p in accordance with FIG. 3.

[0081] It is aimed to solve the system of equations numerically in orderto predict the behavior of the system, that is to say values are to becalculated for the unknown vector x(t) at one or more points in time t.Use is made for this purpose of known methods of numerical solution (notillustrated here) that run as a computer program on a computer system(not shown here).

[0082] According to the invention, methods for providing errorinformation relating to inconsistencies in the system of equations areexecuted for the actual solution of the system of equations.

[0083] The dependence matrix A specified in FIG. 4 is determined in step1 of the method according to the invention, all non-vanishing elementsof the dependence matrix A being denoted by a star (“*”) in FIG. 4.

[0084] The element A(1,1) picked out by way of example is set to anessentially arbitrary non-vanishing value “*”, because the first row off, that is to say f₁ (t,x(t), . . . , x ^((k))(t)), is a function of thefirst element of x, that is to say of x ₁(t). The same holds for theelements A(1,2), A(1,3), A(2,3) and A(3,3).

[0085] The element A(3,1) picked out by way of example is set to thevalue “0”, the third element of f, that is to say f₃ (t,x(t),{dot over(x)}(t), . . . , x ^((k))(t)),is independent of the first element of x,that is to say of x ₁(t) and of derivatives of the first element of x,that is to say x ₁ ^((S))(t). The same holds for the elements A(2,1),A(3,2) and A(2,2).

[0086] In step 2 of the method according to the invention, the firststep is to determine the transversals T, specified in FIG. 4, of thedependence matrix A with the aim of determining row ranks and columnranks of the dependence matrix A specified in FIG. 4.

[0087] The row rank {2, 3} of the dependence matrix A is found by themethod for calculating row ranks (“minimally structured singular subsetsof equations”) specified in C. C. Pantelides “The consistentinitialization of differential algebraic systems”. The set Z of the rowranks, found in step 2, of the dependence matrix A is specified in FIG.4.

[0088] The row rank {l, 2} of the transpose of the dependence matrix Ais found by the method, specified in C.C. Pantelides, “The consistentinitialization of differential-algebraic systems”, for calculating rowranks applied to the transpose of the dependence matrix A. According tothe invention, this row rank of the transpose of the dependence matrix Ais understood as the column rank of the dependence matrix A. The set Sof the column ranks, found in step 2, of the dependence matrix A isspecified in FIG. 4. The error information specified in FIG. 5 is outputin step 3.

[0089] In accordance with the invention, there is not even an attempt topredict the behavior of the basic system using numerical means, becausein this case errors will occur if a simulation is at all possible.However, the modeling of the system and the system itself need to bechecked once again. This saves valuable computing time on the computersystem (not shown here). Error locating is substantially simplified bythe knowledge of the error information output in step 3.

[0090] A third exemplary embodiment in accordance with the inventionrelates to the technical system shown in FIG. 6, whose behavior can bedescribed by the numerical solution of the system of equations

f ₁(x ₁(t),x ₂(t),x ₃(t), p ₁ , p ₂ , p ₃)=0

[0091] of the form

f ₂(x ₁(t),x ₂(t),x ₃(t), p ₁ , p ₂ , p ₃)=0

f ₃(x ₁(t),x ₂(t),x ₃(t), p ₁ , p ₂ , p ₃)=0

[0092] having the functions f1, f2 and f3 and having the parametervector p in accordance with FIG. 3.

[0093] It is aimed to solve the system of equations numerically in orderto predict the behavior of the system, that is to say values are to becalculated for the unknown vector x(t) at one or more points in time t.Use is made for this purpose of known methods of numerical solution thatrun as a computer program on a computer system.

[0094] Error information relating to inconsistencies in the relevantsystem of equations is provided in this case as follows according to theinvention.

[0095] The solutions of the relevant system of equations are quiescentstates or “operating points” or “DC solutions” of the electric networkspecified in FIG. 6 and which comprises the following network elements(“components”):

[0096] a linear resistor with resistance value R between the nodes 1 and2 of the network,

[0097] a linear capacitor with a capacitance C1 between the nodes 1 and0 of the network, and

[0098] a linear capacitor with a capacitance C2 between the nodes 2 and0 of the network.

[0099] This results in the components of the parameter vector p,specified in FIG. 3, being according to the values C1, 1/R, C2.

[0100] The components x1(t), x2(t) and x3(t) of x(t) correspond to thefollowing variables of the network specified in FIG. 6:

[0101] x1(t) corresponds to the voltage between the nodes 1 and 0,

[0102] x2(t) corresponds to the voltage between the nodes 2 and 0, and

[0103] x3(t) corresponds to the voltage between the nodes 1 and 2.

[0104] The first equation of the system of equations, that is to say,the equation

f 1(x 1(t), x 2(t), x 3(t), p 1, p 2, p 3)=0

[0105] is the Kirchhoff voltage equation for the mesh comprising all thethree network elements of the network from FIG. 6.

[0106] The second equation of the system of equations, that is to say,the equation

f 2(x 1(t), x 2(t), x 3(t), p 1, p 2, p 3)=0

[0107] is the Kirchhoff current equation for the nodes 1 of the circuitin FIG. 6.

[0108] The third equation of the system of equations, that is to say,the equation

f 3(x 1(t), x 2(t), x 3(t), p 1, p 2, p 3)=0

[0109] is the Kirchhoff current equation for the nodes 2 of the circuitin FIG. 6.

[0110] The equation significance list G specified in FIG. 7 and thecomponent significance list K specified in FIG. 7 are designed inaccordance with the invention.

[0111] The set Z specified in FIG. 4, of the determined row ranks, andthe set S, specified in FIG. 4 of the determined column ranks aredetermined as in the preceding exemplary embodiments.

[0112] The error information specified in FIG. 8 is output in step 3 ofthe method developed further in accordance with the invention, use beingmade of the equation significance list G and the component significancelist K from FIG. 7.

[0113] It may be seen from the error information specified in FIG. 8that the Kirchhoff current equations relating to the nodes 1 and 2 ofthe network from FIG. 6 are linearly dependent on one another, and thatthe two voltages between nodes 1 and 0 and between nodes 2 and 0 of thenetwork from FIG. 6 for quiescent states of this network cannot beuniquely determined.

[0114] In accordance with the invention, there is not even an attempt topredict the behavior of the basic system using numerical means, becausein this case errors will occur if a simulation is at all possible.However, the modeling of the system and the system itself need to bechecked once again. This saves valuable computing time on the computersystem. Error locating is substantially simplified by the knowledge ofthe error information output in step 3.

[0115] The term “computer-readable medium” as used herein should beunderstood in its broadest sense. That is, it includes, at the least,any kind of computer memory such as floppy disks, hard disks, CD-ROMS,flash ROMs, non-volatile and volatile ROM and RAM, and memory cards, aswell as carrier signals for distance communication.

I claim:
 1. A method for predicting the behavior of a technical systemstarting from prescribed system properties, boundary conditions, and/orstarting from prescribed influences on the system, the method whichcomprises the following steps: providing a computer system; providing asystem of equations describing the technical system, the equationshaving the form f(t,x(t), {dot over (x)}(t), . . . , x ^((k))(t),p)=0,and including: $\begin{matrix}{{{f_{1}\left( {t,{\underset{\_}{x}(t)},{\underset{\_}{\overset{.}{x}}(t)},\ldots \quad,{{\underset{\_}{x}}^{(k)}(t)},\underset{\_}{p}} \right)} = 0},} \\{{{f_{2}\left( {t,{\underset{\_}{x}(t)},{\underset{\_}{\overset{.}{x}}(t)},\ldots \quad,{{\underset{\_}{x}}^{(k)}(t)},\underset{\_}{p}} \right)} \eqsim 0},} \\\vdots \\{{{f_{n}\left( {t,{\underset{\_}{x}(t)},{\underset{\_}{\overset{.}{x}}(t)},\ldots \quad,{{\underset{\_}{x}}^{(k)}(t)},\underset{\_}{p}} \right)} = 0},}\end{matrix}$

wherein x(t) and derivatives {dot over (x)}(t), . . . , x ^((k)) thereofin each case have m elements, and p is a parameter vector that can occurin the system of equations; and executing a test method for providingerror information relating to inconsistencies in the system of equationson the computer system, the test method having the following steps 1 to3: step 1: setting up a dependence matrix A with m columns and n rows,and setting an element A to A(i,j)≠0 when an i^(th) row of f definedwith f _(i)(t,x(t), {dot over (x)}(t), . . . , x ^((k))(t),) is afunction of a) a j^(th) element of x expressed as x _(j) ^((S))(t); orb) one of the derivatives of the j^(th) element of x defined as x _(j)^((S))(t); and otherwise setting the element A(i, j)=0; step 2:determining a set of row ranks each having the numbers of those rows ofthe dependence matrix A that are mutually dependent and determining aset of column ranks of the dependence matrix A each having the numbersof those columns of the dependence matrix A that are mutually dependentif such row ranks and/or column ranks are present; and step 3:outputting error information containing the numbers contained in eachrow rank determined in step 2 and in each column rank determined in step2.
 2. The method according to claim 1, which further comprises: prior toexecuting step 1: applying an equation significance list of length n inwhich each equation of the system of equations is assigned at least oneof an equation number and an item of equation text information; andapplying a component significance list of length m in which eachcomponent of a solution vector x is assigned at least one of a componentnumber and an item of component text information; in step 3, outputtingat least one of the equation number and the item of equation textinformation in accordance with the equation significance list instead ofoutputting the numbers contained in each row rank; and in step 3,outputting at least one of the component number and the item ofcomponent text information in accordance with the component significancelist instead of outputting the numbers contained in each column rank. 3.A computer program with computer-executable instructions for executing amethod for numerical simulation of a technical system according toclaim
 1. 4. A computer program with computer-executable instructions forexecuting a method for numerical simulation of a technical systemaccording to claim
 2. 5. A computer system programmed to execute amethod for numerical simulation of a technical system according toclaim
 1. 6. A computer system programmed to execute a method fornumerical simulation of a technical system according to claim
 2. 7. Acomputer-readable medium having computer-executable instructions forperforming the method according to claim
 1. 8. A computer-relatedmethod, comprising: downloading a computer program product or a computerprogram with computer-executable instructions for executing a method fornumerical simulation of a technical system according to claim 1 from anelectronic-data network onto a computer connected to the data network.9. The method according to claim 8, wherein the electronic data networkis the Internet.
 10. A computer program product and computer program forproviding error information relating to inconsistencies in a system ofequations to be solved with the aid of a computer program running on acomputer or with the aid of an analog computer, which product isdesigned in such a way that it is possible to execute a method forproviding error information relating to inconsistencies in a system ofequations to be solved with the aid of a computer program running on acomputer or with the aid of an analog computer and is of the formf(t,x(t), {dot over (x)}(t), . . . , x ^((k))(t),p)=0, that is to say$\begin{matrix}{{{f_{1}\left( {t,{\underset{\_}{x}(t)},{\underset{\_}{\overset{.}{x}}(t)},\ldots \quad,{{\underset{\_}{x}}^{(k)}(t)},\underset{\_}{p}} \right)} = 0},} \\{{{f_{2}\left( {t,{\underset{\_}{x}(t)},{\underset{\_}{\overset{.}{x}}(t)},\ldots \quad,{{\underset{\_}{x}}^{(k)}(t)},\underset{\_}{p}} \right)} = \underset{\_}{0}},} \\\vdots \\{{{f_{n}\left( {t,{\underset{\_}{x}(t)},{\underset{\_}{\overset{.}{x}}(t)},\ldots \quad,{{\underset{\_}{x}}^{(k)}(t)},\underset{\_}{p}} \right)} = \underset{\_}{0}},}\end{matrix}$

where x(t) and derivatives {dot over (x)}(t), . . . , x ^((k))(t)thereof respectively have m elements, and p is a parameter vector thatcan occur in the system of equations; and the method comprising thefollowing steps: step 1: setting up a dependence matrix A with m columnsand n rows, and setting an element A to A(i,j)≠0 when an i^(th) row of f0 defined with f _(i)(t,x(t), {dot over (x)}(t), . . . , x ^((k))(t),)is a function of a) a j^(th) element of x expressed as x _(j)(t); or b)one of the derivatives of the j^(th) element of x defined as x _(j)^((S))(t); and otherwise setting the element A(i,j)=0; step 2:determining a set of row ranks each having the numbers of those rows ofthe dependence matrix A that are mutually dependent and determining aset of column ranks of the dependence matrix A each having the numbersof those columns of the dependence matrix A that are mutually dependentif such row ranks and/or column ranks are present; and step 3:outputting error information containing the numbers contained in eachrow rank determined in step 2 and in each column rank determined in step2.
 11. The computer program product and computer program according toclaim 10, wherein the method for providing error information relating toinconsistencies further comprises: prior to executing step 1: applyingan equation significance list of length n in which each equation of thesystem of equations is assigned at least one of an equation number andan item of equation text information; and applying a componentsignificance list of length m in which each component of a solutionvector x is assigned at least one of a component number and an item ofcomponent text information; in step 3, outputting at least one of theequation number and the item of equation text information in accordancewith the equation significance list instead of outputting the numberscontained in each row rank; and in step 3, outputting at least one ofthe component number and the item of component text information inaccordance with the component significance list instead of outputtingthe numbers contained in each column rank.
 12. A data carrier with acomputer program product or computer program according to claim
 10. 13.A data carrier with a computer program product or computer programaccording to claim
 11. 14. The computer program product or computerprogram according to claim 10 carried in a computer-readable medium. 15.The computer program product or computer program according to claim 11carried in a computer-readable medium.
 16. A computer system programmedto execute the instructions contained in the computer program product orcomputer program according to claim
 10. 17. A computer system programmedto execute the instructions contained in the computer program product orcomputer program according to claim
 11. 18. A method for predicting abehavior of a system, which comprises programming the computer systemaccording to claim 16 with at least one of prescribed system properties,boundary conditions, and starting from prescribed influences on thesystem, and predicting the behavior of the system by executing thecomputer program.
 19. A computer-related method, comprising: downloadinga computer program product or a computer program according to claim 10from an electronic data network onto a computer connected to the datanetwork.
 20. The method according to claim 19, wherein the electronicdata network is the Internet.